System and Method for Maneuver Plan for Satellites Flying in Proximity

ABSTRACT

A technique to assist guidance techniques for a free-flying inspection vehicle for inspecting a host satellite. The method solves analytically in closed form for relative motion about a circular primary for solutions that are non-drifting, i.e., the orbital periods of the two vehicles are equal, computes the impulsive maneuvers in the primary radial and cross-track directions, and parameterizes these maneuvers and obtain solutions that satisfy constraints, for example collision avoidance or direction of coverage, or optimize quantities, such as time or fuel usage. Apocentral coordinates and a set of four relative orbital parameters are used. The method separates the change in relative velocity (maneuvers) into radial and crosstrack components and uses a waypoint technique to plan the maneuvers.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a nonprovisional under 35 USC 119(e) of, and claims the benefit of, U.S. Provisional Application 61/700,982 filed on Sep. 14, 2012, the entire disclosure of which is incorporated herein by reference.

BACKGROUND

1. Technical Field

The application is related to methods and systems for inspecting satellites with inspection vehicles that travel a path around the satellite to be inspected.

2. Related Technology

Artificial satellites in orbit around the earth can occasionally have problems that require a visual inspection to detect and diagnose. A small vehicle can be sent to move in a path around the satellite to take photographs and inspect the larger satellite.

A satellite (the secondary) circumnavigating another satellite (the primary) in order to inspect it for possible damage or failure will be guided by two goals: first, to avoid collisions with the main satellite, and second, to pass through certain directions (or perhaps, all directions) from the primary from which it is desirable to have a view; a stuck deployable might be imaged for diagnosis and repair on the ground, or perhaps an all-over surface inspection is necessary. If the primary has a protuberance like an antenna or solar panel, and the inspection needs to be at a close distance (on the order of meters), then it may be necessary to have a complicated trajectory in order to meet both conditions. Techniques for planning trajectories for orbital maneuvering have been used successfully for many years, but these techniques do not generally deal with obstacle avoidance. In the last decade, however, spacecraft proximity operations has increased in importance, and consequently techniques for safely operating spacecraft in close proximity to each other have been developed.

Some of these techniques have taken classical astrodynamics as their starting point. Such algorithms typically aim to produce either natural motion trajectories (governed primarily by orbital dynamics) or forced motion trajectories (governed primarily by on-board spacecraft thrusting) that maintain enough distance between the co-orbiting bodies that collisions are impossible. Implicit in this approach is that the co-orbiting bodies' geometry is unimportant; essentially, the bodies are treated as spheres that circumscribe the real geometry of the spacecraft in question. A collision-free trajectory is then one in which the circumscribing spheres do not intersect. This approach has many advantages.

In contrast, the terrestrial robotics community has treated trajectory planning very differently. Robotic trajectory planning is typically concerned with finding collision-free trajectories in highly cluttered or confined environments; one canonical trajectory planning problem in terrestrial robotics entails a mobile vehicle operating inside an office building. Trajectory planning for spacecraft proximity operations based on the classic terrestrial robotics approach thus have the ability to plan much closer maneuvers than those based on classic astrodynamics.

Some approaches are described in U.S. Patent Application Publication No. 2007/0179685 to Milam et al. and 2009/0132105 to Paluszek et al. Relative motion about a primary in circular orbit in terms of centered relative orbital objects is described in L. M. Healy and C. G. Henshaw, “Passively safe relative motion trajectories for on-orbit inspection”, AAS 10-265, pp. 2439-2458, the entire disclosure of which is incorporated herein by reference.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a cross sectional view of a satellite with protrusions and obstacles to be avoided during an inspection by an inspection or secondary vehicle.

FIG. 2 illustrates motion of the inspection vehicle from an initial point to a maneuver waypoint, and subsequently applying thrust to move to a different point.

FIG. 3 is illustrates the initial waypoint, the target waypoint, and the maneuver waypoint and their associated orbits.

FIG. 4 illustrates a reference frame in which to describe the secondary's motion, having a radial component (î axis), an along-track component perpendicular to the radial and in the orbital plane (ĵ axis), and a component perpendicular to the orbital plane parallel to the angular momentum ({circumflex over (k)} axis).

FIG. 5 is a plot of delta-v versus phase at maneuver point for various phases and velocities.

FIG. 6 is a plot of maneuver time versus phase at maneuver point.

DETAILED DESCRIPTION OF THE INVENTION

The system and method described herein determines the maneuvers needed to keep an inspector vehicle close to the host without colliding while being able to inspect the desired faces (directions from the center) of the host. The purpose of this might be to inspect the antenna that will not deploy on a satellite, and potentially to repair the broken antenna with the secondary or inspection vehicle. Additional information is disclosed in “Formation maneuver planning for collision avoidance and direction coverage”, AAS/AIAA Space Flight Mechanics Meeting, AAS 12-102, (2012), the entire disclosure of which is incorporated herein by reference.

FIG. 1 illustrates a host spacecraft 100 having an irregular shape. The protrusions 102, 104 on the spacecraft present obstacles that must be avoided during the inspection maneuver. These protrusions can be solar panels, antenna, or other components. The method has three goals: to maneuver the inspection vehicle close to the host vehicle, to avoid colliding with obstacles, and to minimize fuel usage.

The method uses apocentral coordinates and a set of four constants of the motion that parametrize the relative orbit. The method solves a periodic three-point boundary value problem relative motion about a circular orbit without perturbations. This finds, given a pair of points relative to the primary body, an orbit that connects them. A trajectory consists of a sequence of such natural motion segments connected at points at which an impulsive thrust is executed, and the value of that thrust can be computed by taking the vector difference of the velocities at these common points.

The maneuvers have only radial and cross-track components; by having no in-track component, the motion stays periodic, i.e., the spacecraft have identical semimajor axes and thus orbital periods. Two satellites orbiting closely with the same semimajor axis will stay together over an extended time if no perturbations or other forces are acting on them; the figure of motion of one relative to another will be an ellipse or degenerate ellipse centered somewhere along the track of the primary. Any propulsion in the in-track (inertial velocity) direction will cause a change in the semimajor axis, so unless executed identically on both satellites will cause the formation to come apart.

Therefore, any maneuvers executed with the intent of maintaining the stability of the formation (but not necessarily the configuration) should have components only in the radial and cross-primary-plane directions; if propulsion fails or is incorrect, the formation will still stay together.

This strategy reduces the free variables in the optimal and feasible trajectory planning problem, and thereby solutions are easier to obtain. Furthermore, collision avoidance is much easier to confirm geometrically. A possible disadvantage is that by excluding possible solutions, an optimal solution might be missed. However, the advantages are believed to outweigh the disadvantages, and in realistic relative orbit schemes, a solution found with this method with these constraints will be preferred for practical purposes.

A purely radial maneuver will shift the relative motion ellipse forward or backward in the in-track direction, and change the scale of the ellipse. This affects two of the four parameters, the in-track center y_(c), and the scale, represented by the semiminor axis of the relative ellipse projected to the orbital plane of the primary b. A cross-track maneuver will change the relative orbital plane, as defined by the amplitude ratio η and the phase difference Ξ.

A potential obstacle on the initial path can be identified by looking at a cross section of the primary in the relative orbital plane. There are two ways to do find a safe path if there is an obstacle: enlarge the orbit, which preserves the plane, or change the plane, which will change the eccentricity and size of the ellipse and also the cross-section of the primary. The change in plane may change the direction of view. Within the constraints of collision avoidance and desired viewing direction, there many possible trajectories, as defined by location of maneuver points and value of delta-vat those points. Therefore, additional objective functions such as fuel consumption (proportional to the magnitude of delta-v) or transfer time maybe optimized or at least considered.

This approach is different than the customary approach to this problem. Instead of applying a complete optimal path planner to the full dynamics problem, it is chosen to decompose the problem by developing a deep understanding of the dynamics. This knowledge is applied to both give an intuitive understanding to possible trajectories, and to reduce the number of degrees of freedom, so that when an optimizer is finally applied, it is far more likely that it will converge quickly to a satisfactory result.

The work presented here helps to solve problems of relative orbital guidance for proximity operations. Scharf, Hadaegh, and Ploen provide an overview of the subject in the context of space robotics, which is our intended application. Lovell and Tragesser address relative orbital guidance for different applications, and some of the quantities used herein have analogues in their work. Mullins used the Hill's state-transition matrix to solve the free-time boundary value problem including drift and drag for circular reference orbits. Jiang, Li, Baoyin, and Gao generalize his work and solve the free-time boundary value problem for elliptical orbits by solving the Lambert problem for each vehicle and then linearizing the time equation, using a Newton-Raphson method to solve the problem approximately. The present approach described herein builds in periodic motion as a constraint and gives an exact analytic solution in closed form for relative motion about a circular orbit. Richards, Schouwenaars, How, and Feron use relative motion dynamics to formulate a mixed integer linear programming approach which provides minimum delta-vcollision-free trajectories by numerical optimization. Henshaw and Sanner used an optimal variational technique and the full gravitating-body orbit dynamics; while fully general, in practice, solutions are difficult to obtain due to small basins of convergence. The entire disclosure of each of these documents is incorporated herein by reference.

Composition of Orbit Segments

FIG. 2 and FIG. 3 illustrates the orbital motion of the inspection vehicle along its orbital path 200 from an initial point 201 to a maneuver waypoint 202. At the maneuvering waypoint 202, thrust is applied to move the vehicle to the target waypoint 203 along a path 204, after which the vehicle can continue along its new orbital path 205, moving further to a final point 206.

Turning first to FIGS. 2 and 3, it is a goal to find a path plan of orbital motion of the secondary relative to the primary satisfying certain constraints such as waypoints or the equivalent through which the orbit must pass. These could also be lines from the primary center, for example, to specify a direction over which the secondary should pass to satisfy the need for some observation. An obstacle may be indicated as something the path should avoid by constructing waypoints around it that guide the secondary on a safe path.

A path plan will consist of alternating propagation without maneuvers and impulsive maneuvers with components in the radial and/or cross-track directions. This can be done in a passively safe way, so that if a maneuver fails to happen on schedule, there will be no collision, and a new maneuver to achieve the desired goal can be computed, if the propulsion becomes operational again. The trajectory is safe if the relative orbital ellipse clears the cross-section of the primary sliced by the relative orbit plane, with a band added for the radius of the secondary, plus a margin of safety.

Mathematically, the propagation is a two-point periodic boundary value problem. That is, we define two points on the orbit, and then solve for the orbit between them, represented by the four parameters b, y_(c), η, and Ξ. The position (and velocity) at any point in time between the points may then be determined. That way, it is possible to confirm that obstacles from the primary shape are avoided by insuring that the position on a radial line in a given direction exceeds that of the primary's perimeter in that direction.

The propagations, during which no external force is applied, are interrupted by maneuvers. These maneuvers are presumed to be impulsive, or instantaneous, so that the secondary changes relative velocity at that instant, but not its position. The delta-v (the change in velocity, which is proportional to fuel used) that is used to effect the maneuver may have a radial (î) component or cross-track ({circumflex over (k)}) component; no component in the primary in-track direction (ĵ) is permitted because that would induce a secular separation of the spacecraft, unless counteracted. We shall show how to find the magnitude of these components from the values of the relative position vectors at the waypoints: the departure waypoint, and the target waypoint. Once the maneuver has been computed, it can be confirmed with propagation that the trajectory in fact does reach the target; at the same time obstacle avoidance can be confirmed. This chain of alternating propagation and maneuvers is represented schematically as shown in FIG. 1 and with the orbital motions in FIG. 2.

Apocentral Coordinates

The technique to be presented relies heavily on apocentral coordinates and the relative orbital parameters we introduced previously in L. M. Healy and C. G. Henshaw, “Passively safe relative motion trajectories for on-orbit inspection”, AAS 10-265, pp. 2439-2458, (2010) the entire disclosure of which is incorporated herein by reference. A brief review follows.

Referring next to FIG. 4, the primary defines the RSW reference frame in which to describe the secondary's motion; it consists of the radial component (î axis), the along-track component perpendicular to the radial and in the orbital plane (ĵ axis), and perpendicular to the orbital plane parallel to the angular momentum ({circumflex over (k)} axis). Motion described in these coordinates are well understood in the approximation that they are small and that no perturbations are acting with the Hill's or Clohessy-Wiltshire equations. The components in the three directions are labeled x, y, and z respectively.

If it is assumed that the primary is in a circular orbit, Hill's equations have an analytic form as a function of time t:

$\begin{matrix} {\mspace{79mu} {{x(t)} = {{4\; x_{0}} + \frac{2\; {\overset{.}{y}}_{0}}{n} + {\frac{{\overset{.}{x}}_{0}}{n}\sin \; n\; t} - {\left( {\frac{2{\overset{.}{y}}_{0}}{n} + {3\; x_{0}}} \right)\cos \; n\; t}}}} & (1) \\ {{y(t)} = {{\frac{2\; {\overset{.}{x}}_{0}}{n}\cos \; n\; t} + {\left( {{6\; x_{0}} + \frac{4\; {\overset{.}{y}}_{0}}{n}} \right)\sin \; n\; t} - {\left( {{6\; n\; x_{0}} + {3\; {\overset{.}{y}}_{0}}} \right)t} - \frac{2\; {\overset{.}{x}}_{0}}{n} + y_{0}}} & (2) \\ {\mspace{79mu} {{z(t)} = {{z_{0}\cos \; n\; t} + {\frac{{\overset{.}{z}}_{0}}{n}\sin \; n\; t}}}} & (3) \\ {\mspace{79mu} {{\overset{.}{x}(t)} = {{{\overset{.}{x}}_{0}\cos \; n\; t} + {\left( {{3n\; x_{0}} + {2\; {\overset{.}{y}}_{0}}} \right)\sin \; n\; t}}}} & (4) \\ {\mspace{79mu} {{\overset{.}{y}(t)} = {{\left( {{6\; n\; x_{0}} + {4\; {\overset{.}{y}}_{0}}} \right)\cos \; n\; t} - {2\; {\overset{.}{x}}_{0}\sin \; n\; t} - \left( {{6n\; x_{0}} + {3{\overset{.}{y}}_{0}}} \right)}}} & (5) \\ {\mspace{79mu} {{\overset{.}{z}(t)} = {{{\overset{.}{z}}_{0}\cos \; n\; t} - {z_{0}n\; \sin \; n\; t}}}} & (6) \end{matrix}$

Initial conditions (t=0) are indicated with the subscript 0 (e.g. x₀, y₀) and the primary's mean motion is designated by n. We consider only the periodic (non-drifting) case where x_(c)=4x+2{dot over (y)}/n=0, so that the secular term in time in y(t) is zero.

The motion described is that of an ellipse, which lies in the relative orbital plane. The orientation, size, and eccentricity of the ellipse are given by the geometric relative orbital elements. This ellipse defines a right-hand orthogonal coordinate system we call the apocentral coordinates (by analogy to the perifocal coordinates of gravitating body orbit mechanics) in which the origin is the center of the ellipse, the major axis (apse) provides the first reference axis, the perpendicular in the relative orbital plane provides the second axis, and normal to that plane provides the third.

The center of the ellipse is not necessarily at the primary; in fact any displacement of it in-track is a valid relative orbit. The displacement

$\begin{matrix} {y_{c} = {y - \frac{2\overset{.}{x}}{n}}} & (7) \end{matrix}$

is a constant of motion. In general we will use dimensionless quantities to do this analysis, so instead of lengths, we will have ratios. That means that some quantity must establish the length scale. The in-plane semiminor axis, a constant of motion, is related to the Cartesian coordinates

$\begin{matrix} {b = \sqrt{x^{2} + \left( \frac{y - y_{c}}{2} \right)^{2}}} & (8) \end{matrix}$

and will serve as the scale. These two scalars define the motion in the primary orbital plane by setting the location and size.

In the third dimension, orthogonal to the primary plane along the {circumflex over (k)} axis, we define two more constants of the motion, the amplitude ratio

$\begin{matrix} {\eta = {\frac{1}{b}\sqrt{z^{2} + \left( \frac{\overset{.}{z}}{n} \right)^{2}}}} & (9) \end{matrix}$

and the phase difference

Ξ=arctan(nz,ż)−arctan(−3nx−2{dot over (y)},{dot over (x)}),  (10)

using the two-argument arctangent to obtain the correct quadrant. These define the relative orbital plane. Note however that Healy et al., AAS 10-265, used the notation A for this quantity.

The state of a spacecraft in periodic relative motion about another in an inertial circular orbit can be completely described with the four scalars b, y_(c), η, and Ξ. All the remaining geometric terms are defined in terms of these four.

The transformation from the RSW frame to the apocentral coordinates is given by a rotation (η,Ξ) and a translation defined by y_(c)=y−2{dot over (x)}/n, which is a constant of motion,

r _(apoc)=(η,Ξ)(r−r _(c))  [EQNO]

r _(c) =ĵy _(c)

(η,Ξ)=

_(align)

_(RSW)  (13)

with rotations

RSW  ( η , Ξ ) = [ 0 2 X η   sin   Ξ X X Z 2  η 2  sin   Ξ   cos   Ξ XZ 4  η   cos   Ξ XZ 2  η   cos   Ξ Z η   sin   Ξ Z - 2 Z ]   and ( 14 ) align  ( η , Ξ ) = [ cos   ϖ sin   ϖ 0 - sin   ϖ cos   ϖ 0 0 0 1 ] . ( 15 )

The magnitudes X and Z are defined by

X=√{square root over (4+η² sin² Ξ)},  (16)

Z=√{square root over (4+η²(1+3 cos² Ξ))}.  (17)

The pitch ω is the angle in the relative orbital plane between the apse line and the local horizontal (ĵ−{circumflex over (k)} plane),

$\begin{matrix} {\varpi = {\arctan \left( \frac{Z\; \sin \; \tau_{\max}}{{X^{2}\cos \; \tau_{\max}} + {\eta^{2}\sin \; \Xi \; \cos \; \Xi \; \sin \; \tau_{\max}}} \right)}} & (18) \end{matrix}$

with the value of the phase at the extremum

$\begin{matrix} {\tau_{ext} = {\frac{1}{2}{\arctan \left( {{\eta^{2}\sin \; 2\Xi},{3 - {\eta^{2}\cos \; 2\Xi}}} \right)}}} & (19) \\ {{\tau_{\max} = {\tau_{ext} + {s\frac{\pi}{2}}}},} & (20) \end{matrix}$

where s=0 if the extremum is a maximum

η² cos 2(Ξ+τ_(ext))<3 cos 2τ_(ext)  (21)

and ±1 if a minimum, so that −π/2≦ ω≦π/2.

The semimajor and semiminor axis of the ellipse are expressed in terms of the b, η, and Ξ as well,

$\begin{matrix} {{A = {\frac{b}{X}\left( {{\left\lbrack {{4\; \cos \; \tau_{\max}} + {\eta^{2}\sin \; \Xi \; {\sin \left( {\Xi + \tau_{\max}} \right)}}} \right\rbrack \cos \; \varpi} + {Z\; \sin \; \tau_{\max}\sin \; \varpi}} \right)}},} & (22) \\ {B = {\frac{b}{X}{\left( {{\left\lbrack {{4\; \sin \; \tau_{\max}} - {\eta^{2}\sin \; \Xi \; {\cos \left( {\Xi + \tau_{\max}} \right)}}} \right\rbrack \sin \; \varpi} + {Z\; \cos \; \tau_{\max}\cos \; \varpi}} \right).}}} & (23) \end{matrix}$

The phase angle on orbit is measured with

τ=arctan(−3nx−2{dot over (y)},{dot over (x)})  (24)

which is zero τ=0 when the secondary is in the local horizontal in front of the primary and increases linearly in time at the rate of the primary's mean motion n. When τ=τ_(max), the secondary is the furthest from the primary

FIG. 4 is a plot of relative motion in the relative orbital plane with η=2.0 and Ξ=137.5. When the secondary is at _(max), then τ=τ_(max) and θ=0. In particular, FIG. 4 shows relative orbital motion in its own plane, showing rotation from horizontal {circumflex over (X)} to maximum radius r. The +{circumflex over (X)} axis lies in the direction of motion (ĵ), and the Ŷ axis is perpendicular to it in the relative orbital plane (and so is not necessarily radially upward).

There are two important differences between the inertial and relative motion. First, the ellipse describing the motion can be anywhere along the in-track direction, as reflected in the parameter y_(c), but the inertial orbital ellipse must have its focus at the gravitational center of the gravitating body. The second is that any combination of values of inertial orbital elements is possible, but the geometric relative orbital elements are constrained in ways explained in Healy et al., AAS 10-265, (2010) although they were called “centered relative orbital elements” in that document. As a result, on the one hand, the complete freedom of inertial motion (for example, orbits of any inclination can be of any eccentricity) is absent, but the ability to shift the center is a significant bonus.

The Three-Point Periodic Boundary Value Problem

Problem Statement:

The problem to be solved is the following: given two position vectors in time order i, j, in the RSW coordinates of the secondary relative to the primary, with the primary in a circular orbit, find the following:

(a) the relative orbit that connects the points and returns to the first point after one orbital period, as expressed by the four parameters b, y_(c), η, and Ξ,

(b) the phase change proportional to the elapsed time between the points on this relative orbit, Δτ=nΔt, and

(c) the relative velocity vectors at these points on this relative orbit.

This is the three-point periodic boundary value problem for relative motion about a circular orbit; the three points are 0, 1, and 0 again one orbital period later. It is an analogue for relative motion of the famous Lambert problem, described in John E. Prussing and Bruce A. Conway. Orbital mechanics. Oxford University Press, New York, 1993. In the Lambert problem, there is one degree of freedom: the semimajor axis may be varied (within limits) to get different elapsed times, orbits, and endpoint velocities, for the same pair of points. Here however, as we shallsee, there is no such freedom; once the points are specified, there is a single relative orbit connecting them and the time elapsed between 0 and 1 may be found.

In the apocentral coordinate frame, the relative motion is described by a centered axis-aligned ellipse with semimajor axis length A and semiminor axis length B, with phase angle θ=τ−τ_(max) moving uniformly in time ({dot over (θ)}=n),

$\begin{matrix} {r_{apoc} = {\begin{bmatrix} {A\; \cos \; \theta} \\ {B\; \sin \; \theta} \\ 0 \end{bmatrix}_{apoc}.}} & (25) \end{matrix}$

Motion of the secondary is confined to the first two coordinates; that is, they describe the figure plane. To get the Cartesian position, we use the transpose of

, taking advantage of the fact that a rotation is an orthogonal transformation,

r=

^(T)(η,Ξ)r _(apoc) +r _(c),  (26)

using the offset vector giving the displacement of the center relative to the primary Equation (12),

$\begin{matrix} {{r_{c} = \begin{bmatrix} 0 \\ y_{c} \\ 0 \end{bmatrix}},} & (27) \end{matrix}$

with the secondary periodic (x_(c)=0), this is constant in time. The velocity is computed by differentiation of Equation (25) and Equation (26), noting that the apocentral transformation is independent of time,

r . apoc = n  [ - A   sin   θ B   cos   θ 0 ] apoc   and ( 28 ) r . = T  ( η , Ξ )  r . apoc . ( 29 )

Thus the position or velocity at any time can be found it if the four parameters and time, as represented by θ, are known. Next, compute these parameters given position vectors at two different times.

Scale and Offset

Assuming know position vectors r_(i) and r_(j) are known at two different times, one can find the elements within the primary orbital plane, the scale b, offset y_(c), and the phase θ on the ellipse. The scale (in-plane semiminor axis) b may be computed with Equation (8) for either point. To find the offset y_(c), compute for each point and set them equal:

$\begin{matrix} {{b = {\sqrt{x_{0}^{2} + {\frac{1}{4}\left( {y_{0} - y_{c}} \right)^{2}}} = \sqrt{x_{1}^{2} + {\frac{1}{4}\left( {y_{1} - y_{c}} \right)^{2}}}}},} & (30) \end{matrix}$

since b is constant over the initial orbit. If we square this expression and rearrange,

$\begin{matrix} {{{\frac{1}{4}\left\lbrack {y_{0}^{2} - {2\; y_{0}y_{c}} - \left( {y_{1}^{2} - {2\; y_{1}y_{c}}} \right)} \right\rbrack} = {x_{1}^{2} - x_{0}^{2}}},} & (31) \end{matrix}$

or solving for y_(c),

$\begin{matrix} {y_{c} = {\frac{{4\left( {x_{1}^{2} - x_{0}^{2}} \right)} + y_{1}^{2} - y_{0}^{2}}{2\left( {y_{1} - y_{0}} \right)}.}} & (32) \end{matrix}$

Note that no solution is available if y₀=y₁.

Relative Orbital Plane

The two known points, r_(i) and r_(j), define the relative orbital plane. This plane is most conveniently specified by its normal N,

$\begin{matrix} {{N = {{- {{sgn}(\xi)}}\frac{\left( {r_{i} - r_{c}} \right) \times \left( {r_{j} - r_{c}} \right)}{{\left( {r_{i} - r_{c}} \right) \times \left( {r_{j} - r_{c}} \right)}}}},} & (33) \end{matrix}$

with ξ=[(₀−_(c))×(₁−_(c))]·k. This vector is normalized, although the magnitude doesn't matter. The third component of this vector must be negative because all relative orbits revolve around the primary in the opposite sense of how the primary revolves around the earth; there is no “short way” or “long way” choice as there is in the Lambert problem.

In terms of the relative amplitude η and phase difference Ξ, the normal to the relative orbital plane is given by

$\begin{matrix} {Z = {\begin{bmatrix} {2\eta \; \cos \; \Xi} \\ {\eta \; \sin \; \Xi} \\ {- 2} \end{bmatrix}.}} & (34) \end{matrix}$

Therefore the relative amplitude and phase difference may be computed from a normal N of any magnitude,

$\begin{matrix} {\Theta = {\arctan \left( {{2N_{j}},N_{i}} \right)}} & (35) \\ {\eta = {\sqrt{\frac{N_{i}^{2} + {4N_{j}^{2}}}{N_{k}^{2}}}.}} & (36) \end{matrix}$

the values don't depend on magnitude of this vector N=|N|. If the two points when projected into the primary orbital plane are colinear, a solution is not possible. This is the reason for the admonition above to avoid Δτ=π or multiples. It is possible to pick the solution connection two vectors pointing in the opposite directions from the center by declaring the plane that they have in common. If the two projected centered vectors on the same line and point in the opposite direction, a solution is physically possible only if the components in the k direction have the same magnitude with the opposite sign. However, the cross product will be zero.

From the two parameters η and Ξ, pitch ω and the apocentral transformation, the semimajor axis A, the semiminor axis B, and τ_(max) can be calculated. Also available are the eccentricity e, the slant σ (the angle between the relative and primary orbital planes), the elevation of the node

(the angle from the local horizontal plane to the intersection of the relative and primary orbital planes), though they are not needed for the immediate calculation. At either the initial or final point, the phase θ is computed in the next section, and from that the velocity Equation (28). The gives the complete relative orbital state.

In picking a final point, to pick the {circumflex over (k)} component of final position vector 1, we might pick values that have a certain relative orbital plane. As with the i, j components, the value might be picked to satisfy certain constraints (like collision avoidance) or optimize a parameter, such as time or delta-v usage. This will be discussed in the maneuvers section.

Phase

The phase θ=τ−τ_(max) is proportional to the time elapsed since the secondary passed the major axis (FIG. 2) or apse; the constant of proportionality is the primary mean motion n. It is analogous to the mean anomaly ininertial orbit mechanics. Compute the apocentral vector using the apocentral transformation Equation (11)

r _(apoc)=

(η,Ξ)(r−r _(c)),  (37)

and then solve for the angle by using the first two components of apocentral position vector from Equation (25),

θ=arctan(Ay _(apoc) ,Bx _(apoc)),  (38)

using the two-argument arctangent. The time elapsed for the secondary to travel between the points is easily computed from the difference in phase,

$\begin{matrix} {{\Delta \; t} = {\frac{\theta_{1} - \theta_{0}}{n}.}} & (39) \end{matrix}$

For this formula to produce the correct result, the phase angle should be computed so that it does not decrease with time. This means that it may be necessary to add or subtract multiples of 2π to the arctangent result.

With the phase at the endpoints θ_(i), θ_(j) known, the velocity at those points {dot over (r)}_(i), {dot over (r)}_(j) (or at any other points on the orbit) may be computed with Equation (29).

Summary of Steps

The three-point periodic boundary value problem takes as input two relative position vectors and the mean motion of the primary n. The points must be non-colinear, not both in the local horizontal plane, and not have the same in-track position. The steps to solve are as follows:

-   -   1. Compute the four parameters y_(c) (Equation (32)), b from         either position vector using (Equation (8)), η and Ξ (Equation         (34)).     -   2. Find X, Z (Equation (15)), τ_(max) (Equation (20)), ω         (Equation (18)), A (Equation (22)), B (Equation (23)).     -   3. Find the apocentral transformation         (Equation (10)) using (Equation (14)) and (Equation (15)).     -   4. Find the apocentral position vector (Equation (37)) for         either point.     -   5. Find the orbit phase θ (Equation (38)) for both points.     -   6. Find the elapsed time Δt (Equation (39)) to travel between         the points.     -   7. Find the relative velocity at any point from the phase,         (Equation (28)) and (Equation (29)).

For this algorithm to work, the two given points must not be colinear relative to the center because Equation (33) will not solve, nor may they have common in-track components, because Equation (32) will not solve. They must not both be entirely in the local horizontal (j−k) plane, because the only relative orbit whose normal is entirely in the primary orbital plane is a degenerate one that passes through the primary, oscillating on either side on the k axis.

This is a closed-form analytic solution to the two point boundary value problem for closed (non-drifting) relative motion about a circular orbit without perturbations. The analogous problem in gravitating body orbit mechanics is the Lambert problem. The Lambert problem has a degree of freedom that this problem does not; it is usually expressed as the freedom to select the semimajor axis, which correspondingly affects the time. Moreover, two directions are possible in the Lambert problem, the short way and the long way, and there is no choice here. In solving the Lambert problem for a fixed time, an iteration is necessary to converge on the correct semimajor axis. No such iteration is necessary here, and there is no choice of the time. The freedom to change the time and delta-v is gained in the selection of intermediate waypoints.

EXAMPLE

Suppose we wish the relative orbit to pass between two points,

$\begin{matrix} {{r_{0} = {\begin{bmatrix} {- 1} & 000 \\ 3 & 000 \\ 1 & 500 \end{bmatrix}m}},{r_{2} = {\begin{bmatrix} 1 & 400 \\ {- 0} & 500 \\ {- 2} & 200 \end{bmatrix}{m.}}}} & (40) \end{matrix}$

If the primary orbits the earth at an altitude of 981.32 km, the mean motion will be [0.001]rads. Using Equation (32), we compute the offset y_(c)=[0.7014]m. The plane normal is then computed from Equation (33),

$\begin{matrix} {N = {\begin{bmatrix} {- 0} & 8498 \\ {- 2} & {611 \times 10^{- 2}} \\ {- 0} & 5265 \end{bmatrix}.}} & (41) \end{matrix}$

Since the third component is negative, the sign need not be changed. This is an indication that the angle between the vectors is less than 180 going clockwise in the −k direction. From this, we compute Equation (34) the phase difference Ξ=−176.5 and the relative amplitude η=1.617. The apocentral rotation matrix is

$\begin{matrix} {{= \begin{bmatrix} 0 & 1628 & 0 & 9369 & {- 0} & 3093 \\ 0 & 5014 & {- 0} & 3485 & {- 0} & 7919 \\ {- 0} & 8498 & {- 2} & {611 \times 10^{- 2}} & {- 0} & 5265 \end{bmatrix}},} & (42) \end{matrix}$

so that the initial position vector₀ in apocentral coordinates Equation (37),

$\begin{matrix} {\,_{0}{= {\begin{bmatrix} 1 & 527 \\ {- 2} & 490 \\ 0 & 0 \end{bmatrix}_{apoc}.}}} & (43) \end{matrix}$

The ellipse axis half-lengths are A=[1.414]m and B=[1.321]m, so the initial phase Equation (38) is θ₀=41.03 and the final phase is θ₂=113.2, which corresponds to an elapsed time Equation (39) of 44 m 52.170 s at the altitude given. The velocity at the initial time t₀ is then computed Equation (29)

$\begin{matrix} {{\overset{.}{r}}_{0} = {\begin{bmatrix} 1 & {149 \times 10^{- 3}} \\ 2 & {000 \times 10^{- 3}} \\ {- 1} & {954 \times 10^{- 3}} \end{bmatrix}{{ms}.}}} & (44) \end{matrix}$

The results are easily checked by using the complete state (position and velocity) initial conditions in the Hill's equations solutions for a circular orbit Equation (3) and propagating the computed time to find the given final position r₂.

Maneuvers

Effect of Maneuver Components in RSW Coordinate Directions

The effects of each component can be considered an impulsive maneuver on the parameters.

If delta-vpoints in the radial (i) direction, the component of motion perpendicular to the primary orbital plane will remain the same; if the motion previously was in the primary orbital plane, it will stay that way after the maneuver. The parameters referring to motion in the primary orbital plane, b and y_(c), will change. As a consequence, the phase difference Ξ may also change. Since there is no change in the cross-track motion and b has changed, the relative amplitude η will change. The radial center x_(c) is unchanged, so the motion remains periodic. The change in the offset y_(c) Equation (7) repositions the center and the ellipse forward or backward along the direction of motion. The change in the scale b Equation (8) and the center happen in such a way that the projection of the point of impulse on the primary orbital plane is common to both projected ellipses. The maneuver causes an instantaneous change in τ Equation (24). If there is a component of delta-v in the cross-track direction, then c and φ will change. The illustration in the next section will clarify this motion.

A delta-v in the in-track direction j changes x_(c) so that drift is induced which results in motion that is not closed, and can be complicated. Because of the complexity of analysis with drifting, we leave this aside. Assume that x_(c)=0 before and after the maneuver.

A delta-v in the cross-track direction {circumflex over (k)} changes cross-plane motion independently of the in-plane motion. Therefore, the parameters η and Ξ will change.

Delta-v Effects on GROE

Any maneuver of the secondary can be analyzed as components in the RSW frame. We shall look at the components in the i, j, and k directions separately, covering the following elements:

a. radial delta-v gives change of b, y_(c), ψ

b. cross-track delta-vgives change of c, φ

c. combine change η, Ξ

d. implies change of σ,

e. others: A, e, ω follow as consequence of η, Ξ

f. Inverse direction from GROE differences back to changes in velocity components

Defining the relative orbital plane can be used to find the η and Ξ by Equation (34), then from there, go up the list or down the list to find how other quantities change. General delta-v: higher η means more expensive changes in phase Ξ, but gives better coverage of sides. We seek to understand the relation between impulsive maneuvers in the radial and cross-track directions, and the change in parameters. We approach this problem in the following sections by first looking at the motion projected in the primary orbital plane, and then expanding that to full three-dimensional motion.

Radial Delta-v

The projection of the relative orbit into the primary orbital plane can be considered to be parametrized by just two quantities: the amplitude b, and the offset y_(c). For a single impulsive radial maneuver, there will be two sets of such parameters, the “before” indicated with a superscript “−” and after with superscript “−”; e.g., b⁻, b⁺ are the initial and final scales, respectively. From these values, one can compute the radial delta-v needed to effect the maneuver. There are effects on the other orbital elements which we will compute. Finally, how a given radial delta-v changes these parameters is considered.

In this section, find the radial delta-v given the change in offset y_(c), and that in turn is computed knowing two points of the segment (x₁, y_(i)) and (x_(j), y_(j)) through which the trajectory passes, provided the in-track positions are different, y_(i)≠y_(j).

From Healy et al. 2010, using Equations (11a) and (10b) with x_(c)=0,

$\begin{matrix} \begin{matrix} {\overset{.}{x} = {{nb}\; {\cos \left( {\psi + {n\; t}} \right)}}} \\ {= {n\frac{y - y_{c}}{2}}} \end{matrix} & (45) \end{matrix}$

with n the mean motion of the primary Therefore, the radial delta-v is related to the change in offset Δy_(c)=y_(c) ⁺−y_(c) ⁻,

$\begin{matrix} \begin{matrix} {{\Delta \; \overset{.}{x}} = {\frac{1}{2}{n\left( {y_{c}^{+} - y_{c}^{-}} \right)}}} \\ {= {\frac{1}{2}n\; \Delta \; {y_{c}.}}} \end{matrix} & (46) \end{matrix}$

To find the offset before and after a maneuver, we need only have a pair of points for each orbit; for an impulsive maneuver, they can have the relative position at the point of maneuver in common.

Cross-Track delta-v

We now introduce cross-track delta-v and with it, changes to the cross-track amplitude c. The amplitude ratio η is obviously affected by this amplitude, and consequently the relative ellipse semimajor and semiminor axes A, B. It affects the slant σ and pitch ω. From Healy et al., 2010, Equation (13f), the cross-track velocity is dependent on c=bη and Ξ:

ż=nbη cos(Ξ+τ).  (47)

Therefore, if the new and old values of these quantities are known, cross-track delta-v can be computed as,

Δż=nb[η ⁺ cos(Ξ⁺+τ)−η⁻ cos(Ξ⁻+τ)].  (48)

Note that this assumes no radial component to the delta-v; if there is such a component, then both b and τ would change as well.

Changing a Single Orbital Parameter

There are four parameters through which one can completely describe a periodic relative orbit: b, y_(c), η, and Ξ. By analogy with gravitating body orbit mechanics where the orbital maneuvers first studied are those that change only one orbital element, we may here look at those that change only one of these parameters. Where the initial and final relative orbits have intersection points, only a single impulsive thrust is necessary to accomplish the maneuver. Where they do not, at least two separate thrusts are required, and there will be one or more intermediate transfer orbits that the secondary must be on for some period of time. Here, one can consider a single transfer orbit for each case, because if an orbit can be found that connects any two points, it will not be necessary to search for an intersection point if there is one.

A conceptually simple way to avoid an obstacle while maintaining the same directional views of the primary is to resize the orbit but maintain its center and relative orbital plane. This will require two maneuvers, like a Hohmann transfer. Start with two known points r₀ and r₁ to define the relative orbit, and solve the boundary value problem to find the velocity when the secondary is at the first point; we will call this {dot over (r)}₀ ⁻ as it will be the velocity immediately before maneuvering at this point. With the computed values of y_(c), η, and Ξ retained, we can rescale b by a scalar α, say by doubling. There is now a point r′₀=αr₀ on the same radial line from the center as r₀; we compute the rest of the quantities from step 4.6 onward in the summary Then pick some phase change for the transfer orbit Δθ_(t) to propagate on this orbit from r₀; call the new point r₁ and its velocity {dot over (r)}₁ ⁺. Now we solve the boundary value problem again, this time between r₀ and r₁. The computed relative velocity at r₀ on this orbit we will call {dot over (r)}₀ ⁺, and propagating this transfer orbit, the velocity at r₁ is {dot over (r)}₁ ⁻. Finally, the delta-vs are

Δ{dot over (r)} ₀ ={dot over (r)} ₀ ⁺ −{dot over (r)} ₀ ⁻  (49)

Δ{dot over (r)} ₁ ={dot over (r)} ₁ ⁺ −{dot over (r)} ₁ ⁻  (50)

Δv=|{dot over (r)} ₀ |+|{dot over (r)} ₁|;  (51)

and the time of transfer may be computed from Equation (39).

For one example, suppose it is desired to double the size of the orbit, with α=2.

If the initial position is

${r_{0} = {\begin{bmatrix} {- 1.} & 000 \\ 3. & 000 \\ 1. & 500 \end{bmatrix}m}},$

In this initial orbit, the scale is b=[1.008]m when doubled, we will have b=[2.016]m. At the point r₀, the position and velocity on the initial orbit are

$\begin{matrix} {{r_{0} = {\begin{bmatrix} {- 1.} & 000 \\ 3. & 000 \\ 1. & 500 \end{bmatrix}m}},{{\overset{.}{r}}_{0} = {\begin{bmatrix} 0. & 1250 \\ 2. & 000 \\ {- 1.} & 125 \end{bmatrix}{mms}}},} & (53) \end{matrix}$

and a maneuver is executed based on a destination position₁ found by propagating the final orbit by a phase Δθ_(t)=90 from the rescaled point r₀,

$\begin{matrix} {{r_{1} = {\begin{bmatrix} 0. & 2500 \\ 6. & 750 \\ {- 2.} & 250 \end{bmatrix}m}},} & (54) \end{matrix}$

with a delta-v computed by taking the difference of the velocities on the two orbits,

$\begin{matrix} \begin{matrix} {{\Delta {\overset{.}{r}}_{0}} = {{\overset{.}{r}}_{0}^{+} - {\overset{.}{r}}_{0}^{-}}} \\ {= {\begin{bmatrix} {- 0.} & 6875 \\ 2. & 000 \\ {- 1.} & 687 \end{bmatrix} - \begin{bmatrix} 0. & 1250 \\ 2. & 000 \\ {- 1.} & 125 \end{bmatrix}}} \\ {= {\begin{bmatrix} {- 0.} & 8125 \\ 2. & 0 \\ {- 0.} & 5625 \end{bmatrix}{{mms}.}}} \end{matrix} & (55) \end{matrix}$

After a phase change of Δθ=136.4 (39 m 40.580 s at our standard altitude) on the transfer orbit, we execute another maneuver

$\begin{matrix} \begin{matrix} {{\Delta {\overset{.}{r}}_{1}} = {{\overset{.}{r}}_{1}^{+} - {\overset{.}{r}}_{1}^{-}}} \\ {= {\begin{bmatrix} 2. & 000 \\ {- 0.} & 5000 \\ {- 3.} & 000 \end{bmatrix} - \begin{bmatrix} 1. & 188 \\ {- 0.} & 5000 \\ 0. & 1875 \end{bmatrix}}} \\ {= {\begin{bmatrix} 0. & 8125 \\ 0. & 0 \\ {- 3.} & 187 \end{bmatrix}{mms}}} \end{matrix} & (56) \end{matrix}$

at the point.

The total delta-v is Δv=[4.278×10⁻³]ms and the total elapsed time for the transfer is 39 m 40.580 s.

Changing the Center

Analogous to resizing, we may change the in-track center by a fixed displacement δ. Again, we start with r₀ and r₁ to define the relative orbit, and solve the boundary value problem. We retain the computed value of the scale b and substitute for the offset y_(c)=y′_(c)+δ, and keep η, and Ξ. The calculations proceed as before, resulting in delta-vs and time of transfer.

Changing the Amplitude Ratio

The amplitude ratio η is changed solely by a delta-v in the cross track ({circumflex over (k)}) direction, if we keep b constant. Since the elevation of the node is independent of η, both the initial and final orbits will cross the primary orbital plane at the same point, so we may perform a single impulsive thrust at that point.

In the RSW coordinate frame, the scaled position and velocity of the secondary are given in Healy et al., AAS AAS 10-265, Equations (16) and (18)

$\begin{matrix} {{\frac{r}{b} = \begin{bmatrix} {\sin \; \tau} \\ {2\cos \; \tau} \\ {{\eta sin}\left( {\Theta + \tau} \right)} \end{bmatrix}},{\frac{\overset{.}{r}}{bn} = {\begin{bmatrix} {\cos \; \tau} \\ {{- 2}\sin \; \tau} \\ {{\eta cos}\left( {\Theta + \tau} \right)} \end{bmatrix}.}}} & (57) \end{matrix}$

If for some integer m, τ=mπ−Ξ, then the third component of is zero. In that case, the third component of the velocity is (−1)^(m)nbη. While an amplitude ratio η change will change the plane normal, from it can be seen that it does not change the phase difference Ξ. Therefore, if for some value of Δη the delta-v

Δv _(z)=(−1)^(m) nbΔη  (58)

is executed when τ=mπ−Ξ for an integer m, then the amplitude ratio will change by Δη, and b, y_(c), and Ξ will remain constant.

Changing the Phase Difference

Finally, changing the phase difference Ξ also changes the plane. We can do it in two maneuvers. Again, we start with ₀ and ₁ to define the relative orbit, and solve the boundary value problem. We retain the computed value of the scale b, offset y_(c), and η, and replace the value of Ξ. The calculations proceed as before, resulting in delta-vs and time of transfer.

Feasible and Optimal Trajectories

Goals

There are two goals we consider when designing the trajectory of a secondary acting as an inspector of the primary: collision avoidance and coverage. Coverage of the primary is the set of directions from the primary through which the secondary passes. The goal may be imaging of a single part of the surface of the primary, or imaging all over the surface.

Collision avoidance means that the trajectory does not pass through any parts of the primary If the primary and secondary are spheres, this is easy: any relative orbit whose minimum distance from the center is greater than the sum of the radii of the primary and secondary is safe. If they are not spheres, we can insure safety by imagining a safety sphere enveloping each that has a radius at least as large as the largest distance from the center of every point on the spacecraft. However, if we wish the secondary to come closer to the primary, say for inspection purposes, that procedure won't work.

A trajectory is passively safe if the relative orbit does not intersect with the host. In the absence of any maneuver then, it will stay on the safe trajectory. To maximize safety, we should minimize the number of impulsive maneuvers, on the premise that the greatest chance for failure is at a maneuver. There are two possible failure scenarios at a maneuver: the more likely in our assumption is that nothing happens; there is no delta-vas desired. Because of the design of the non-maneuvering relative orbit, this event (or non-event) is harmless: the secondary keeps on its safe relative orbit, though perhaps without achieving an imaging goal. The other failure scenario is that the actual delta-v is not the commanded delta-v; a misfire. This could well put the secondary on a collision course, and there is little from a trajectory design perspective we can do to prevent this. The best we can do is minimize risk by minimizing the number of maneuvers.

We have assumed here that there is no in-track maneuvering, as this results in a orbit for the secondary that has different orbital period than the primary, and they separate secularly (actually, relative motion is periodic with the synodic period). A way to keep the pair together is to counteract the in-track maneuver with an opposite maneuver some time later. However, the trajectory may not be passively safe.

Referring again to FIG. 1, one approach to collision avoidance for very close motion is to find the cross section of the secondary on the relative orbital plane. The relative orbital plane can be thought of slicing through space, and through the primary, so that we can see a relative orbit around a cross-section of the primary. If the maneuver will preserve the relative orbital plane, then compute a plane-preserving shift and/or scale and test whether the ellipse intersects the primary. If it will change the plane, then a new cross-section of the primary will be needed, and the new ellipse should be tested for intersection. An ellipse and rectangle may be tested for intersection by a method such as the one described by Ratschek and Rokne. An actual cross section, such as could be obtained from a CAD model, would likely not be a simple rectangle, but could be decomposed into rectangles, with an intersection test performed on each one. If the plane is changed (i.e. one or both of η and Ξ) for an attitude-stabilized primary, then a new cross-section will have to be taken to determine the obstacles to be avoided.

If the primary is rotating in the RSW frame, planning a trajectory for direction and for collision avoidance for very close motion will be much harder. There is possibly some benefit to simplifying the trajectory; for example, if the primary is spinning about its velocity vector j, then if the secondary stays in the primary orbital plane, it will see all directions on the primary (presuming the spin and orbital periods are not commensurable).

EXAMPLE

In a previously described example, in which the relative orbit was doubled, the free parameter, Δθ_(t), was chosen to be 90. This parameter can be adjusted and the trajectory recalculated. This has been done, and the results of the delta-v and the time plotted in FIG. 5 and FIG. 6, respectively. If there are obstacles on the transfer orbit, a new trajectory may avoid them. Time on the transfer orbit and/or fuel used may be an issue as well.

Note that a proposed maneuver can be evaluated for fuel efficiency by examining the many points that form FIG. 5. For example, the peaks in the curve for Δv are the points that require the most thrust. Therefore, it would be wise to avoid the regions around the peaks, and select a phase at the maneuver point at which the Δv is low. Similarly, the time required to complete a maneuver for each phase at the maneuver point is shown in FIG. 6. The two plots can be used together to select an appropriate phase at the maneuver point.

The method and system described herein can plan a trajectory for relative motion where the primary is in a circular orbit, the secondary in a periodic (non-drifting) orbit relative to it, there are no perturbations acting, and the linear approximation (as used to derive Hill's equations) holds. Maneuver points at which external forces are applied impulsively are alternated propagation with no external forces. The maneuvers include only radial and cross-track components (there is no in-track component), so that throughout the trajectory, the orbit is periodic. Between every pair of points, we solve the three-point periodic boundary value problem for relative motion, the solution for which we have presented here based on our previous work. This solution is unique and an analytical function of its arguments. Once the velocities are computed at the maneuver points, the delta-vs are easily obtained by computing a vector difference.

The waypoints may be chosen so that the secondary avoids collisions, so that it has desired directional properties relative to the primary, so that fuel usage may be minimized, or so that transfer time is a desired value. If, for example, it is desired that the secondary follow a certain trajectory relative to the primary, the waypoints may be chosen freely such that the trajectory satisfies those constraints. For example, in doubling the size of the relative orbit, the target point for the second maneuver may be varied over its orbit, and the resulting transfer trajectories have very different delta-vs, time of transfer, and potential for collision.

Any desired trajectory can be achieved with a sufficiently fine filling of waypoints. For example, suppose that an inspector needed to travel along a long flat surface, staying approximately a constant distance away from that surface. A natural orbital motion would be an arc, and therefore not uniformly distant. Bisecting the length of the surface with a point at the right distance would give two arcs, better, but likely still not enough. Bisecting each of those with points would produce better results, and successive bisections would eventually yield an emulation of a straight line with small arcs sufficient to achieve the requirement of near-constant distance. In this analysis, we assume the only force on the two spacecraft is the planetary central gravitation.

Clearly, differential perturbations will change these results somewhat, and it is believed that the algorithms presented here can be generalized to accommodate them. Likewise, the circularity of the primary orbit and linearity approximation may prove significant in some circumstances when generalizing this technique.

This method has several differences and advantages over previous approaches.

The techniques that are based on classical astrodynamics have both advantages and disadvantages. Starting as it does with classical astrodynamics techniques, it results in trajectories that explicitly take orbital dynamics into account, and do not require the coorbiting spacecraft to expend fuel to travel in un-natural ways such as following straight lines. It also tends to result in solutions that are amenable to implementation with existing spacecraft propulsion systems, which are typically considered impulsive.

Unfortunately, this approach also has significant limitations. Foremost among these is that it does not treat the case where the two spacecraft must operate at ranges closer than their circumscribing spheres allow. For instance, a GEO satellite may have twin solar panels as long as 25 meters, which implies that its circumscribing sphere is at least 50 meters in diameter; this implies that classic approaches to proximity operations trajectory planning cannot produce solutions that allow proximity operations closer than 50 meters for such a spacecraft.

The terrestrial robotics approach also has several important disadvantages when applied to spacecraft. Primary among these is that the terrestrial robotics community often (although certainly not always) ignores system dynamics; instead, it is assumed that the robot is capable of accurately tracking any given trajectory, even a trajectory that is only piecewise linear, so closely that dynamic effects can be ignored. Adapting a classical terrestrial robotics trajectory planning approach for co-orbiting spacecraft would require ignoring orbital dynamics and assuming that the inspection spacecraft is assumed to have enough control authority and on-board fuel to perform essentially any delta-v.

Thus, neither of these classes of solutions is entirely satisfactory. Techniques which can generate paths that satisfy orbital dynamic constraints and allow very close approach distances are needed.

The present method derives a new way of specifying the motions of one spacecraft relative to another, using as our inspiration the classic orbital elements. In this method, the mathematical space in which relative satellite motion can be intuitively understood, and relatively complex geometric obstacle constraints can be easily expressed. By adapting terrestrial trajectory planning techniques in such a space, the techniques combine the fuel efficiency of classic astrodynamics with the close approach distances allowed by classic robotics trajectory planning.

The method described herein can be implemented on a computer, and the thrust vectors are input to the inspection satellite control system, which in turn controls the inspection vehicle velocity and position in space. Feedback can be provided to the computer, including positional information from a communications link with one or both satellites, global positioning satellite data, or other information.

Initial trajectory planning can be accomplished on a ground-based computer, or even on the host satellite computers. It may be necessary to periodically re-calculate the trajectories, in order to compensate for off-course position or to reinspect a particular portion of the host satellite.

Embodiments of the present invention may be described in the general context of computer code or machine-usable instructions, including computer-executable instructions such as program modules, being executed by a computer or other machine, such as a personal data assistant or other handheld device. Generally, program modules including routines, programs, objects, components, data structures, and the like, refer to code that performs particular tasks or implements particular abstract data types. Embodiments of the invention may be practiced in a variety of system configurations, including, but not limited to, handheld devices, consumer electronics, general purpose computers, specialty computing devices, and the like. Embodiments of the invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in association with both local and remote computer storage media including memory storage devices. The computer useable instructions form an interface to allow a computer to react according to a source of input. The instructions cooperate with other code segments to initiate a variety of tasks in response to data received in conjunction with the source of the received data.

Computing devices includes a bus that directly or indirectly couples the following elements: memory, one or more processors, one or more presentation components, input/output (I/O) ports, I/O components, and an illustrative power supply. Bus represents what may be one or more busses (such as an address bus, data bus, or combination thereof). One may consider a presentation component such as a display device to be an I/O component. Also, processors have memory. Categories such as “workstation,” “server,” “laptop,” “hand held device,” etc., as all are contemplated within the scope of the term “computing device.”

Computing devices typically include a variety of computer-readable media. By way of example, and not limitation, computer-readable media may comprise Random Access Memory (RAM); Read Only Memory (ROM); Electronically Erasable Programmable Read Only Memory (EEPROM); flash memory or other memory technologies; CDROM, digital versatile disks (DVD) or other optical or holographic media; magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other tangible physical medium that can be used to encode desired information and be accessed by computing device.

Memory includes non-transitory computer storage media in the form of volatile and/or nonvolatile memory. The memory may be removable, nonremovable, or a combination thereof. Exemplary hardware devices include solid state memory, hard drives, optical disc drives, and the like. Computing device includes one or more processors that read from various entities such as memory or I/O components. Presentation component can present data indications to a user or other device. I/O ports allow computing devices to be logically coupled to other devices including I/O components, some of which may be built in.

Obviously, many modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that the claimed invention may be practiced otherwise than as specifically described, and that the invention is not limited to the preferred embodiments discussed above. 

What is claimed as new and desired to be protected by Letters Patent of the United States is:
 1. A computer implemented method for determining a trajectory from an initial waypoint to a target waypoint for a space vehicle, comprising: computing y_(c), b from either position vector, η and Ξ; finding the X, Z, τ_(max), pitch ω, A, B; finding the apocentral transformation

; finding an apocentral position vector for either point; finding the orbit phase θ for both points; finding the elapsed time Δt to travel between the points; and finding the relative velocity at any point from the phase. 